Question

1. Let "D" and "O" be the demand and supply of a certain good, and P be the price of the same. If the supply (O) and demand (D) functions are represented as follows: O = 50P - 300 \quad D = 150 - 10P a) Calculate the equilibrium price of said good and explain what would happen if the price were higher and if it were lower than $5; $10 respectively. b) Calculate the demand and supply for each of the prices previously analyzed.

Answer 1

To find the equilibrium price, we need to set the supply \( O \) equal to the demand \( D \). The given functions are:

\[ O = 50P - 300 \]
\[ D = 150 - 10P \]

### a) Calculate the equilibrium price.

To find equilibrium, set \( O \) equal to \( D \):

\[
50P - 300 = 150 - 10P
\]

Now, let's solve for \( P \):

1. Add \( 10P \) to both sides:

\[
50P + 10P - 300 = 150
\]

\[
60P - 300 = 150
\]

2. Add \( 300 \) to both sides:

\[
60P = 450
\]

3. Divide by \( 60 \):

\[
P = \frac{450}{60} = 7.5
\]

Thus, the equilibrium price \( P^* = 7.5 \).

### What happens if the price is higher or lower than $7.5:

- **If the price is higher than $7.5** (for example, at \( P = 10 \)):
- Calculate supply:

\[ O = 50(10) - 300 = 500 - 300 = 200 \]

- Calculate demand:

\[ D = 150 - 10(10) = 150 - 100 = 50 \]

- In this case, supply (200) exceeds demand (50), leading to a surplus of goods. Producers will lower their prices to clear this surplus, moving the price down toward equilibrium.

- **If the price is lower than $7.5** (for example, at \( P = 5 \)):
- Calculate supply:

\[ O = 50(5) - 300 = 250 - 300 = -50 \] (indicating no supply)

- Calculate demand:

\[ D = 150 - 10(5) = 150 - 50 = 100 \]

- In this case, demand (100) exceeds supply (0), indicating a shortage of goods. Consumers will compete for the limited supply, driving the price up toward equilibrium.

### b) Calculate the demand and supply for the prices of $5 and $10.

**At \( P = 5 \)**:

- Supply:

\[
O = 50(5) - 300 = 250 - 300 = -50 \quad \text{(no supply)}
\]

- Demand:

\[
D = 150 - 10(5) = 150 - 50 = 100
\]

**At \( P = 10 \)**:

- Supply:

\[
O = 50(10) - 300 = 500 - 300 = 200
\]

- Demand:

\[
D = 150 - 10(10) = 150 - 100 = 50
\]

### Summary of Results:

- At \( P = 5 \):
- **Supply** = 0 (no supply)
- **Demand** = 100

- At \( P = 10 \):
- **Supply** = 200
- **Demand** = 50

The equilibrium price is \( P = 7.5 \), with implications of surplus at higher prices and shortages at lower prices.